Harmonic Motion II Serway 15 2 15 3

Harmonic Motion (II) (Serway 15. 2, 15. 3) • Mass and Spring • Energy in SHM Physics 1 D 03 - Lecture 33 1

Velocity and Acceleration a(t) =- w 2 x(t) Physics 1 D 03 - Lecture 33 2

Energy in SHM Look again at the block & spring We could also write M E = K+U = ½ m(vmax )2 Physics 1 D 03 - Lecture 33 3

Question: Suppose U=½ k. A 2 at t = 0, and T=1 sec. When is U=K during the cycle? (i. e. when do their curves cross? ) Physics 1 D 03 - Lecture 33 4

E U K t U, K oscillate back and forth “out of phase” with each other; the total E is constant. T Physics 1 D 03 - Lecture 33 5

Armed with the equation of motion: and the expressions for energy: We can solve a large range of problems in SHM. Example 1: A 0. 25 kg block is oscillating on a spring with k=4 N/m. At t=0, v=-0. 2 m/s and a=+0. 5 m/s 2. Find its total energy and the equation of motion. Physics 1 D 03 - Lecture 33 6

Solution Physics 1 D 03 - Lecture 33 7

Example 2: A 50. 0 g object connected to a spring k=35. 0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4. 0 cm. Find: a) the total energy of the system b) the speed of the object when the position is 1. 0 cm c) the kinetic energy when x=3. 0 cm d) the potential energy when x=3. 0 cm. Physics 1 D 03 - Lecture 33 8

Example 3: A 200 g mass is attached to a horizontal spring and executes SHM with a period of 0. 25 s. The total energy of the system is 2. 0 J. a) Find the force constant of the spring b) Find the amplitude of the motion Physics 1 D 03 - Lecture 33 9

Example 4: A car having a mass of 1000 kg is driven into a brick wall. The car’s bumper behaves like a spring of constant 5 x 106 N/m and compresses 3. 16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no mechanical energy loss with the wall? Physics 1 D 03 - Lecture 33 10

Example 5: A particle executes SHM with an amplitude of 3. 0 cm. At what position does its speed equal one half of its maximum speed? Physics 1 D 03 - Lecture 33 11

Example 6: A uniform disc is connected to a spring and allowed to roll back and forth without slipping. What is the angular frequency of the oscillations? Physics 1 D 03 - Lecture 33 12